Gaussian Elimination

1 / 90

# Gaussian Elimination - PowerPoint PPT Presentation

Gaussian Elimination. Chemical Engineering Majors Author(s): Autar Kaw http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates. Naïve Gauss Elimination http://numericalmethods.eng.usf.edu. Naïve Gaussian Elimination.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about ' Gaussian Elimination' - nell-campbell

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### Gaussian Elimination

Chemical Engineering Majors

Author(s): Autar Kaw

http://numericalmethods.eng.usf.edu

Transforming Numerical Methods Education for STEM Undergraduates

Naïve Gaussian Elimination

A method to solve simultaneous linear equations of the form [A][X]=[C]

Two steps

1. Forward Elimination

2. Back Substitution

Forward Elimination

The goal of forward elimination is to transform the coefficient matrix into an upper triangular matrix

Forward Elimination

A set of n equations and n unknowns

. .

. .

. .

(n-1) steps of forward elimination

Forward Elimination

Step 1

For Equation 2, divide Equation 1 by and multiply by .

Forward Elimination

Subtract the result from Equation 2.

_________________________________________________

or

Forward Elimination

Repeat this procedure for the remaining equations to reduce the set of equations as

. . .

. . .

. . .

End of Step 1

Forward Elimination

Step 2

Repeat the same procedure for the 3rd term of Equation 3.

. .

. .

. .

End of Step 2

Forward Elimination

At the end of (n-1) Forward Elimination steps, the system of equations will look like

. .

. .

. .

End of Step (n-1)

Back Substitution

Solve each equation starting from the last equation

Example of a system of 3 equations

Back Substitution

THE END

http://numericalmethods.eng.usf.edu

Naïve Gauss EliminationExamplehttp://numericalmethods.eng.usf.edu
Example: Liquid-Liquid Extraction

A liquid-liquid extraction process conducted in the Electrochemical Materials Laboratory involved the extraction of nickel from the aqueous phase into an organic phase. A typical set of experimental data from the laboratory is given below:

Assuming g is the amount of Ni in organic phase and a is the amount of Ni in the aqueous phase, the quadratic interpolant that estimates g is given by

Example: Liquid-Liquid Extraction

The solution for the unknowns x1, x2, and x3 is given by

Find the values of x1, x2, and x3 using Naïve Gauss Elimination. Estimate the amount of nickel in organic phase when 2.3 g/l is in the aqueous phase using quadratic interpolation.

Number of Steps of Forward Elimination

Number of steps of forward elimination is

(n-1)=(3-1)=2

Example: Liquid-Liquid Extraction

Solution

Forward Elimination: Step 1

Yields

Example: Liquid-Liquid Extraction

Forward Elimination: Step 1

Yields

Example: Liquid-Liquid Extraction

Forward Elimination: Step 2

Yields

This is now ready for Back Substitution

Example: Liquid-Liquid Extraction

Back Substitution: Solve for x3 using the third equation

Example: Liquid-Liquid Extraction

Back Substitution: Solve for x2 using the second equation

Example: Liquid-Liquid Extraction

Back Substitution:Solve for x1 using the first equation

Example: Liquid-Liquid Extraction

The solution vector is

The polynomial that passes through the three data points is then

Where g is the amount of nickel in the organic phase and a is the amount of in the aqueous phase.

Example: Liquid-Liquid Extraction

When 2.3 g/l is in the aqueous phase, using quadratic interpolation, the estimated amount of nickel in the organic phase is

THE END

http://numericalmethods.eng.usf.edu

Naïve Gauss EliminationPitfallshttp://numericalmethods.eng.usf.edu
Is division by zero an issue here? YES

Division by zero is a possibility at any step of forward elimination

Pitfall#2. Large Round-off Errors

Solve it on a computer using 6 significant digits with chopping

Pitfall#2. Large Round-off Errors

Solve it on a computer using 5 significant digits with chopping

Is there a way to reduce the round off error?

Avoiding Pitfalls
• Increase the number of significant digits
• Decreases round-off error
• Does not avoid division by zero
Avoiding Pitfalls
• Gaussian Elimination with Partial Pivoting
• Avoids division by zero
• Reduces round off error
THE END

http://numericalmethods.eng.usf.edu

Gauss Elimination with Partial Pivotinghttp://numericalmethods.eng.usf.edu
Pitfalls of Naïve Gauss Elimination
• Possible division by zero
• Large round-off errors
Avoiding Pitfalls
• Increase the number of significant digits
• Decreases round-off error
• Does not avoid division by zero
Avoiding Pitfalls
• Gaussian Elimination with Partial Pivoting
• Avoids division by zero
• Reduces round off error
What is Different About Partial Pivoting?

At the beginning of the kth step of forward elimination, find the maximum of

If the maximum of the values is

in the p th row,

then switch rows p and k.

Example (2nd step of FE)

Which two rows would you switch?

Gaussian Elimination with Partial Pivoting

A method to solve simultaneous linear equations of the form [A][X]=[C]

Two steps

1. Forward Elimination

2. Back Substitution

Forward Elimination

Same as naïve Gauss elimination method except that we switch rows before each of the (n-1) steps of forward elimination.

THE END

http://numericalmethods.eng.usf.edu

Gauss Elimination with Partial PivotingExamplehttp://numericalmethods.eng.usf.edu
Example 2

Solve the following set of equations by Gaussian elimination with partial pivoting

Example 2 Cont.
• Forward Elimination
• Back Substitution
Number of Steps of Forward Elimination

Number of steps of forward elimination is (n-1)=(3-1)=2

Forward Elimination: Step 1

Examine absolute values of first column, first row

and below.

• Largest absolute value is 144 and exists in row 3.
• Switch row 1 and row 3.
Forward Elimination: Step 1 (cont.)

Divide Equation 1 by 144 and

multiply it by 64, .

.

Subtract the result from Equation 2

Substitute new equation for Equation 2

Forward Elimination: Step 1 (cont.)

Divide Equation 1 by 144 and

multiply it by 25, .

.

Subtract the result from Equation 3

Substitute new equation for Equation 3

Forward Elimination: Step 2

Examine absolute values of second column, second row

and below.

• Largest absolute value is 2.917 and exists in row 3.
• Switch row 2 and row 3.
Forward Elimination: Step 2 (cont.)

Divide Equation 2 by 2.917 and

multiply it by 2.667,

.

Subtract the result from Equation 3

Substitute new equation for Equation 3

Back Substitution

Solving for a3

Gauss Elimination with Partial PivotingAnother Examplehttp://numericalmethods.eng.usf.edu
Partial Pivoting: Example

Consider the system of equations

In matrix form

=

Solve using Gaussian Elimination with Partial Pivoting using five significant digits with chopping

Partial Pivoting: Example

Forward Elimination: Step 1

Examining the values of the first column

|10|, |-3|, and |5| or 10, 3, and 5

The largest absolute value is 10, which means, to follow the rules of Partial Pivoting, we switch row1 with row1.

Performing Forward Elimination

Partial Pivoting: Example

Forward Elimination: Step 2

Examining the values of the first column

|-0.001| and |2.5| or 0.0001 and 2.5

The largest absolute value is 2.5, so row 2 is switched with row 3

Performing the row swap

Partial Pivoting: Example

Forward Elimination: Step 2

Performing the Forward Elimination results in:

Partial Pivoting: Example

Back Substitution

Solving the equations through back substitution

Partial Pivoting: Example

Compare the calculated and exact solution

The fact that they are equal is coincidence, but it does illustrate the advantage of Partial Pivoting

THE END

http://numericalmethods.eng.usf.edu

Determinant of a Square MatrixUsing Naïve Gauss EliminationExamplehttp://numericalmethods.eng.usf.edu
Theorem of Determinants

If a multiple of one row of [A]nxn is added or subtracted to another row of [A]nxn to result in [B]nxn then det(A)=det(B)

Theorem of Determinants

The determinant of an upper triangular matrix [A]nxn is given by

Forward Elimination of a Square Matrix

Using forward elimination to transform [A]nxn to an upper triangular matrix, [U]nxn.

Example

Using naïve Gaussian elimination find the determinant of the following square matrix.

Forward Elimination: Step 1

Divide Equation 1 by 25 and

multiply it by 64, .

.

Subtract the result from Equation 2

Substitute new equation for Equation 2

Forward Elimination: Step 1 (cont.)

Divide Equation 1 by 25 and

multiply it by 144, .

.

Subtract the result from Equation 3

Substitute new equation for Equation 3

Forward Elimination: Step 2

Divide Equation 2 by −4.8

and multiply it by −16.8,

.

.

Subtract the result from Equation 3

Substitute new equation for Equation 3

Finding the Determinant

After forward elimination

.

Summary
• Forward Elimination
• Back Substitution
• Pitfalls
• Improvements
• Partial Pivoting
• Determinant of a Matrix